Superconductivity due to Condensation of Monopoles around RCD Strings in SU(2) Gauge Theory

Rajput, B. S.; Kumar, Sandeep

doi:https://doi.org/10.1155/2010/768054

Advances in High Energy Physics

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Volume 2010 | Article ID 768054 | https://doi.org/10.1155/2010/768054

Superconductivity due to Condensation of Monopoles around RCD Strings in SU(2) Gauge Theory

B. S. Rajput¹and Sandeep Kumar²

Academic Editor: George Siopsis

Received17 Sept 2010

Revised24 Nov 2010

Accepted01 Dec 2010

Published24 Jan 2011

Abstract

The study of the condensation of monopoles and the resulting chromomagnetic superconductivity have been undertaken in restricted chromodynamics of SU(2) gauge theory. Constructing the RCD Lagrangian and the partition function for monopoles in terms of string action and the action of the current around the strings, the monopole current in RCD chromo magnetic superconductor has been derived and it has shown that in London' limit the penetration length governs the monopole density around RCD string in chromo magnetic superconductors while with finite (nonzero) coherence length the leading behavior of the monopole density at large distances from the string is controlled by the coherence length and not by the penetration length.

1. Introduction

Quantum chromo dynamics (QCD) is the most favored color gauge theory of strong interaction whereas superconductivity is a remarkable manifestation of quantum mechanics on a truly macroscopic scale. In the process of current understanding of superconductivity, Rajput [1, 2] and Kumar et al. [3, 4] have conceived its hopeful analogy with QCD and demonstrated that the essential features of superconductivity, that is, the Meissner effect and flux quantization, provided the vivid models [5–9] for actual confinement mechanism in QCD. The original ideas explaining color confinement in terms of dual superconductivity of QCD vacuum were proposed by ‘t Hooft and Mandelstam in the series of papers where ‘t Hooft demonstrated [7, 10, 11] that the vacuum of gluodynamics behaves as dual superconductor and the key role in dual superconductor model of QCD is played by Abelian monopoles and Mandelstam [12–14] propounded that the color confinement properties may result from the condensation of magnetic monopoles in QCD vacuum. In a series of papers [15–18] Ezawa and Iwazaki made an attempt to analyze a mechanism of quark confinement by demonstrating that the Yang-Mills vacuum is magnetic superconductor and such a superconducting state is considered to be a condensed state of magnetic monopole. The condensation of magnetic monopole incorporates the state of magnetic superconductivity [19] and the notion of chromo magnetic superconductor where the Meissner effect, confining magnetic field in ordinary superconductivity, would be replaced by the chromoelectric Meissner effect (i.e., the dual Meissner effect) which would confine the color electric flux. As such one conceives the idea of correspondence between quantum chromo dynamic situation and chromomagnetic superconductor. However, the crucial ingredient for condensation in a chromo magnetic superconductor would be the nonAbelian force in contrast to the Abelian ones in ordinary superconductivity. Topologically, a nonAbelian gauge theory is equivalent to a set of Abelian gauge theories supplemented by monopoles [11].The method of Abelian projection is one of the popular approaches to confinement problem, together with dual superconductivity [20, 21] picture, in nonAbelian gauge theories. Monopole condensation mechanism of confinement (together with dual superconductivity) implies that long-range physics is dominated by Abelian degrees of freedom [21] (Abelian dominance).

Evaluating Wilson loops under the influence of the Abelian field due to all monopole currents, monopole dominance has been demonstrated [22, 23]. In the Abelian projection the quarks are the electrically charged particles and, if monopoles are condensed, the dual Abrikasove string carrying electric flux is formed between quark and antiquark. Due to nonzero tension in this string, the quarks are confined by the linear potential. The conjecture that the dual Meissner effect is the color confinement mechanism is realized if we perform Abelian projection in the maximal gauge where the Abelian component of gluon field and Abelian monopoles are found to be dominant [24, 25]. Then the Abelian electric field is squeezed by solenoidal monopole current [26]. The vacuum of gluodynamics behaves as a dual superconductor and the key role in dual superconductor model of QCD is played by Abelian monopole. Therefore an important problem, before studying the vacuum properties of nonAbelian theories, is to abelianize them so as to make contribution of the topological magnetic degrees of freedom to the partition function explicit. To meet this end, a dual gauge theory called restricted chromo dynamics (RCD) (i.e., an Abelian version of nonAbelian QCD) has been constructed out of QCD in SU(2) theory [27–30] by imposing an additional internal symmetry named magnetic symmetry [31–35] which reduces the dynamical degrees of freedom. Attempts have been made [1–4] to establish an analogy between superconductivity and the dynamical breaking of magnetic symmetry, which incorporates the confinement phase in RCD vacuum.

In the present paper this structure of RCD has been used to undertake the study of condensation of monopoles and the resultant chromomagnetic superconductivity in SU(2) gauge theory. The RCD Lagrangian density for monopoles has been derived in magnetic gauge and the resulting partition function has been computed in terms of string action and the action of current around the strings. Using this partition function, the quantum average of Wilson loop for monopoles has been computed and the sources of electric flux (i.e., quarks) running along the trajectory have been introduced with the help of Wilson loop.

The monopole current in RCD chromo magnetic superconductor has been derived in London limit which corresponds to infinitely deep Higgs potential leading to vanishing coherence length. It has been shown that the squared monopole current in RCD chromo magnetic superconductor in the London limit has a maximum at the distance of the order of penetration length and it (the penetration length) governs the monopole density around the string in RCD chromomagnetic superconductor. The monopole current has also been derived in RCD chromo magnetic superconductor with nonzero finite coherence length and it has been shown that the monopole density is nonzero even in the absence of string. It has also been shown that the quantum correction to the squared monopole density is much more than its vacuum expectation value measured far outside the string. It has been demonstrated that in the chromo magnetic superconductors with finite (nonzero) coherence length the quantum corrections to squared monopole density control the leading behavior of the total monopole density in the vicinity of the RCD string. It has also been shown that the leading behavior of the monopole density at large distances from the string is controlled by the coherence length and not by the penetration length.

2. Superconductivity due to Condensation of Monopoles in SU(2) Gauge Theory

Monopole condensation mechanism of confinement, together with dual superconductivity, implies that long-range physics is dominated by Abelian degrees of freedom and the method of Abelian projection (i.e., Abelianization) is one of the popular approaches to the problem of confinement, and hence superconductivity, in nonAbelian gauge theories. Such an Abelianization of QCD may be obtained in the form of restricted chromodynamics (RCD) by imposing an additional internal symmetry named magnetic symmetry [31–35] which reduces the dynamical degrees of freedom. Mathematical foundation of restricted chromodynamics (RCD) is based on the fact that a nonAbelian gauge theory permits some additional internal symmetry, that is, magnetic symmetry [36–39]. Unified space of nonAbelian gauge theory may be thought of as which is dimensional manifold where is 4-dimensional external space and , in general, is the -dimensional internal space, generated by Killing vectors satisfying the conditions where From (3.1) we get the monopole current as Equation (3.25) gives Substituting relations (3.24), (3.21), (3.25), (3.26b), and (3.28) into (3.27), we get where and , summation over repeated index is conventionally involved. Substituting relations (3.24), (3.21) and (3.28) into field equation (3.23), we have where dash devotes derivatives with respect to . At large distance, in view of equations (3.20), we may have where is infinitesimally small at large distance such that Then (3.30) may be written as Substituting into this equation, we have which is modified Bessel’s equation of zero order, with its solution given as where is the modified Bessel’s function of zero order. In the similar manner, the field equation (3.4) may be written into the following form by using relations (3.21) and (3.29); At large distance we may have where .

Then (3.36) reduces to where . Let us substitute into this equation. Then we have which is modified Bessel’s equation of order-one with its solution given by where is modified Bessel’s function of order one. Thus we have

Substituting relations (3.35) and (3.41) into (3.31) and (3.37), we have at large value of , Substituting these results into (3.21) and (3.24) with (3.25), we get the solution of classical field equations (3.2) and (3.23) corresponding to the RCD string with a quark and an antiquark at its ends. The infinitely separated quark and antiquark correspond to an axially symmetric solution of the string. For such a string solution with a lowest nontrivial flux, the coefficient in the solution (3.35) is always equal to one while the coefficient in the solution (3.41) is unity in the BogomoLny limit exactly on the border between the type I and type II superconductors where , that is, coherence length and the penetration length coincide with each other. Thus in RCD close to border, we set besides and then we have

The RCD string is well defined by these solutions.

4. Monopole Density around RCD String

Substituting these relations (3.44) into (3.29), we can find the monopole density in the vicinity of RCD string. To meet this end, let us write the partition function corresponding to RCD Lagrangian of (2.13) in Abelian Higgs Model (AHM) in the following form by using (2.15); This model (AHM) incorporates dual superconductivity and hence confinement as the consequence of monopole condensation since the Higgs-type mechanism arises here.

With this partition function the quantum average of Wilson loop may be written as where the expectation value in r.h.s is calculated in AHM with the operator as the product of ‘t Hooft loop and the Wilson loop ; and Wilson loop given as which creates the particle with electric charge on the world trajectory .

Then the effective electric and magnetic four-current density may be written as follows; In (4.3) the operator is and is the dual field tensor satisfying This operator creates the string spanned by the loop , carrying the flux . In AHM the monopoles are condensed and in its string representation the topological interaction exists in the expectation value of the Wilson loop given by (4.4). In the centre of ANO string the Higgs field vanishes, that is, and the phase is singular on the world sheets of ANO string. Then the measure of the integration over can be written as where is a constant. The integral contains the integration over functions which are singular on two dimensional manifolds. Let us divide the phase into regular and singular parts as where is defined by with , string-position and as the collection of all the closed surfaces.

is the parameterization of string surface and . Then the measure can be decomposed as Let us use these relations to find the monopole density in the vicinity of RCD string for vanishing and nonvanishing coherence lengths respectively in the following subsections.

4.1. For Zero Coherence Length

From (2.21), we find the vanishing coherence length in the limit or which corresponds to infinitely deep potential of (2.14). This limit is London limit. Then the RCD Lagrangian of (4.1) in AHM may be written as follows where denotes the Lagrangian density for monopoles. In terms of this Lagrangian, the partition function of (4.1) may be written as follows The string in RCD manifests itself as a singularity in the phase of the Higgs field according to (4.11).

Let us fix the unitary gauge as and make the consequent shift Then the shift in will be where and we have used relations (4.11), (4.16), and (4.13). Substituting shifts (4.16) and (4.17) into (4.14) and integrating over the field , we get where is the string action given as where is the scalar Yukawa propagator. It is the propagator of the scalar particle of mass , that is, with as mass of dual gauge boson .

For closed strings, we have

On the other hand, when the strings are spanned on the current , we have The action of the currents is given as follows by the short-ranged exchange of the dual gauge boson, where is the electric charge of gluon, satisfying the quantization condition The quantum average of Wilson loop can be written here as a sum over strings similar to (4.19), where is given by (4.19). In this equation the sources of electric flux (i.e., quarks) running along the trajectory are introduced with the help of .

Let us place the static quarks at spatial infinities of the axis-. Then the effects of quarks (i.e., boundary effects) are avoided and consequently the second term of the exponential in r.h.s of (4.26) may be ignored. In this case (i.e., infinite static string placed along the third direction) (4.11) reduces to the following form of string current From (3.27), the monopole current may be written as follows in the London limit; When the singular phase corresponds to the string position fixed by (4.11), the Lagrangian in the exponential of (4.14) becomes and then the functional generating the partition function in (4.14) may be written as Then the monopole current in the presence of the string is given by the variational derivative [44]

And the squared monopole density is In the manner analogous to (4.19) and (4.20), the generating functional (4.30) may be written as where string action is given by (4.20).

Substituting this relation for generating functional into (4.32) and evaluating the monopole density, we get the monopole current around the string as For static string, this equation reduces to where with as modified Bessel’s function of zero order. The function given by (3.54), is the propagator for a scalar massive particle in two space-time dimensions. Using (4.34) and (4.36), the explicit form of the nonzero component of the solenoidal current may be written as where is the modified Bessel’s function of order one given in (3.44). Thus the monopoles form a solenoidal current which circulates around the string in transverse directions. This current gives rise to the following squared monopole density;

Substituting the value of from (3.44) into this relation, we find that the squared density of the monopole current, in London limit (where coherence length is zero), has a maximum at the distance of the order of the (i.e., the order of penetration length).

4.2. For Nonzero Coherence Length

When the potential of (2.14) is of finite depth, that is, η is finite then is finite and hence coherence length given by (2.23), is nonzero and finite. Then in the expression (4.38) for squared monopole density in the vicinity of RCD string a term corresponding to quantum vacuum correction is nonzero even in the absence of string. Thus the squared monopole density, in this case, is written as where is given by (4.38) and quantum vacuum correction is given by where we have used (4.36) and regularized the divergent expression by momentum cut off .

Replacing vacuum expectation value of the Higgs field by in relation (4.40) and then substituting it into (4.39), we get For of the order of coherence length the quantum correction to the squared monopole density is much more than the vacuum expectation value measured far outside the string (). Thus the quantum corrections control the leading behavior of the total monopole density in the vicinity of the RCD string.

Using the asymptotic expansions of modified Bessel’s functions in (3.44), and then introducing it into (3.24), we get Then (4.42) may be approximately written as follows [38] at large distances; which shows that the leading behaviors of the monopole density at large distances are controlled by coherence length and not by penetration length .

5. Discussion

The Lagrangian, given by (2.13) for RCD in magnetic gauge in the absence of quarks or any colored objects, establishes an analogy between superconductivity and the dynamical breaking of magnetic symmetry which incorporates the confinement phase in RCD vacuum where the effective potential , given by (2.14), induces the magnetic condensation of vacuum. This gives rise to magnetic super current which screens the magnetic flux and confines the color iso-charges as the result of dual Meissner effect. The confinement of color is due to the spontaneous breaking of magnetic symmetry which yields a nonvanishing magnetically charged Higg’s condensate, where the broken magnetic group is chosen by Abelianization process and hence the magnetic condensation mechanism of confinement in RCD is dominated by Abelian degrees of freedom. Such Abelian dominance in connection with monopole condensation has recently been demonstrated by Bornyakov et al. [45]. The similar result has also been obtained more recently in a dual superconductivity model [21].

In the confinement phase of RCD, the monopoles are condensed under the condition (2.19) where the transition from to is of first order, which leads to the vacuum becoming a chromo magnetic superconductor in the analogy with Higg’s-Ginsburg-Landau theory of superconductivity. Magnetically condensed vacuum is characterized by the presence of two massive modes given by (2.20) and (2.21), respectively, where the mass of scalar mode determines how fast the perturbative vacuum around a color source reaches condensation and the mass of vector mode determines the penetration length of the colored flux. With these two scales of dual gauge boson and monopole field, the coherence length and the penetration length have been constructed by (2.23) in RCD theory. These two lengths coincide at the border between type-I and type-II-superconductors.

The ansatz given by relations (3.10) and (3.11) shows that there is a nontrivial coordinate dependent relative phase between the components of SU(2) doublets. This anastz breaks the originally present global SU(2) symmetry to U(1) and reduces the four-dimensional action of Lagrangian (2.13) to the two dimensional one given by (3.14) with the field equation given by (3.16), (3.17), (3.18), and (3.19). For the special case with the static solution given by (3.21), (3.23) and (3.24) there is neither a relative rotation nor a relative twist between the components of Higg’s field. Relations (3.35) and (3.41) remove the mistakes of the similar relations of Chernodub et al. [44]. Substituting relations (3.44) into (3.21) and (3.24), the solutions of classical field equations (3.1) and (3.2), corresponding to the RCD string with a quark and antiquark at its ends, readily follows. The RCD string is well defined by solutions (3.44) where the monopole current given by (3.27) near the RCD string, is zero at the center of the string and also zero at points far away from the string.

Equation (4.14) gives the partition function in the Eulidean space-time with the RCD Lagrangian density in magnetic gauge given by (4.13). This partition function has been computed in the form given by (4.19) in terms of string action given by (4.20). The action of the current around the strings is given by (4.24) which leads to the quantum average of Wilson loop as given by (4.26) where the source of electric flux (i.e., quarks) running along the trajectory are introduced with the help of Wilson loop.

Equation (4.28) gives the monopole current in London limit which corresponds to infinitely deep Higg’s potential and leads to vanishing coherence length in the chromo magnetic superconductor. Equation (4.31) gives the monopole current in the presence of the string, which leads to squared monopole density given by (4.32). The monopole current given by (4.34) reduces to the components given by (4.35) in terms of propagator (4.36) for a scalar massive particle in two space-time dimensions. Equation (4.37) gives the explicit form of the nonzero component of the solenoidal current which circulates around the string in transverse directions. This current gives rise to the squared monopole current given by (4.38) in London limit (i.e., vanishing coherence length). This squared current has a maximum at the distance of the order of penetration length. Thus in London limit (zero coherence length) the monopole density around the string in RCD is governed by penetration length.

Equation (4.39) shows that for nonzero finite coherence length, the monopole density is nonzero even in the absence of string. Equation (4.42) shows that the quantum correction to the squared monopole density is much more than the vacuum expectation value measured far outside the string. Thus the quantum corrections control the leading behavior of the total monopole density in the vicinity of the RCD string. Equation (4.43) shows that the leading behavior of the monopole density at large distances is controlled by the coherence length and not by the penetration length. This result is in agreement with the numerical result of Bornyakov et al. [45, 46].

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Copyright © 2010 B. S. Rajput and Sandeep Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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